**Part I – Abstract**

**I.2. OUR APPROACH TO THE BEC**

**I.2.2. Loading the dipole trap from an atomic reservoir**

As previously explained, our strategy is based on a very dense and not so cold initial dipole trap, in which evaporation can be very rapid and can produce a BEC in some seconds, outrunning all relevant losses. Therefore, the first issue to be addressed in the analysis of this strategy is the feasibility of the loading of the dipole trap; that is if the dense dipole trap, which is the initial point of the evaporation, is possible to be prepared.

The loading process consists of superimposing the dipole trap with a larger atomic reservoir
which can be maintained for relatively long periods of time (hundreds of µs). During this process,
an atom that is moving in the reservoir can cross several times the trap region, and can end up
trapped in the dipole potential after loosing kinetic energy due to collisions in this region with other
atoms. The fact that this process can be realized for large periods of time, results for important
probabilities for atom transfer and thus for increased loading efficiency. In the realization of the
experiment, the role of the reservoir was played by several types of traps; in particular Compressed
MOT (C-MOT), magnetic trap and Dark SPOT have been used as reservoirs while the experimental
results are presented in the chapter I.4. . Nevertheless, magnetic traps are the largest available traps
with more than 10^{9} atoms at a temperature of ~100 µK directly transferred from a MOT. In the
analysis shown here, we consider a magnetic trap as the reservoir without loss of generality, since
the basic assumptions remain the same and since all collision losses can been taken into account.

The magnitudes that we need to calculate for the loading process are the number of transferred atoms (or the percentage of the atoms that are transferred on those who are initially found in the reservoir) and the temperature of the atoms in the dipole trap once the loading is finished (saturated), as well as an estimation of the loading time. The estimation of the final atom number and temperature is done with thermodynamic arguments. Initially, a first estimation is done without taking into account any collisional losses. After the production of this first results, the most relevant losses are identified and calculated. Finally, a new estimation is given on the atom number, which is the difference of the initial estimation (without losses) minus the losses calculated in this second step.

The description of the transfer process is done in a thermodynamical context. Usually, the process of loading a dipole trap (as well as many other kinds of manipulations) are done adiabatically. However, the authors of [Comp06] show that the temperature rise associated with non-adiabatic (rapid) loading is small for the usual 'reservoir' temperatures (~100 µK), while the experimental simplifications emerging from such an approach are important. Thus, we assume non-adiabatic loading, that is, we assume that the loading starts with a sudden superposition of the dipole potential to the magnetic trap.

Since no adiabaticity is assumed, the loading dynamics are governed by an equation which expresses conservation of energy. In a case where adiabaticity would be assumed, the starting point would be an equation which expresses entropy conservation. Conservation of energy for our system is expressed by Eq. 2.8

*E*T*f**NU**f*=*E** ^{'}*T

*i* (2.8)

where T*i* and T*f *are the initial and final temperatures, U*f** is the final trapping potential depth and N *
the number of trapped atoms, while with E and E' we symbolize the energy of the trapped atoms

after the end and before the start of the loading process. The term NU*f* appears in Eq.2.8 in order to
express the fact that the magnetic and the dipole trap do not have the same potential in their centers,
since the magnetic potential is zero in the trap center r = 0 and the dipole potential in r = ∞. The
energy per atom is given by

*E*T= *F*T*N*_{magn}*k*_{B}*T S*T

*N*_{magn}*k** _{B}* (2.9)

where N*magn* is the initial atom number in the magnetic trap, S(T) and F(T) are the entropy and the
free Helmholtz energy. The entropy is calculated according to

*S*T= *F*T*T*−FT−*T*

2*T N*_{magn}*k*_{B}*T* (2.10)

where with δT is an infinitesimal temperature modification (in this simulation, this temperature
modification equals to 10^{-2} times the initial temperature in the magnetic trap). The free Helmholtz
energy is given by

*F*T=*N*_{magn}*k*_{B}*T*log*DT*−1 (2.11)
where D(T) is the temperature depended phase space density defined as

*DT*= *N*

*Z*T (2.12)

The phase-space density is defined in such way that a value of one corresponds to BEC. This is intuitively close to the picture discussed in the introduction chapter, and implies that the point in which the de Broglie wavelength is in the same order of magnitude as the inter-atomic distances has been reached. Z(T) is the partition function defined here as

*Z*T=* _{dB}*T

^{−3}

### ∫

potential has a dependence on both the laser intensity and the magnetic field gradient, so that*U*=*U*r , I ,∇*B U*_{in}=*U*r,0*,*∇*B* (2.14)
A first characterization of the loading process, without taking into account the possible
losses is done by numerical computation of the temperature from Eq. 2.8 and from integration of the
density from Eq. 2.12 (the integration is done until r equals to 100 times the laser waist). The results
of the are shown in Fig. I.2.3. The initial temperature for the atoms in the reservoir (magnetic trap)
is 150 µK. In this figures, the reservoir loading seems like an excellent scenario, especially because
of the very optimistic phase space densities that it predicts for the loading of small (~50 µm) traps.

These scenarios are not yet realistic, since there are many types of losses have that have not yet been included. Traps of size smaller than 50 µm are not shown here, even if it seems that this model predicts very good results for small traps; the reason is that the image created for tight traps is going to be reversed as soon as we take into consideration the various losses associated with the loading process and in particularly the hydrodynamical regime, and show that traps of small dimensions (<90 µm) are indeed very bad candidates for this loading process.

In order to study the dynamics during the loading process we make some assumptions that
allow us to work with clarity, without loss of generality. One assumption is that the reservoir is
much larger than the dipole trap, so that the reservoir stays unaffected in terms of atom number and
temperature during the process. Thus, in the context of this study, we can write that N*f** << N**i* and T*f *

*~ T**i*, with the last assumption being not so far away from the prediction shown in figure 2.3.

Another simplifying assumption is to consider the factor η, which is the ratio of the dipole trap's potential expressed in units of temperature on the temperature of the atoms, always larger than 4, since this simplifies the analysis and nevertheless this will be the favorable case for the subsequent study of the evaporation process.

The probability to be transferred to or ejected from the dipole trap depends on the energy
change during a collision. For an atom that follows the Boltzmann distribution, the probability to
acquire energy greater than ηk*B**T after a collision is given by the approximate formula*

*p**k*_{B}*T*

E* _{K}* =

*f*

*f*0= 12*e*^{−}

−erf

###

^{}(2.15)

where ^{E}*K*≈3/2*k*_{B}*T* the typical kinetic energy of the atom, erf the error function and
*f* n=

### ∫

*n k**B**T*

∞

###

^{E}

^{'}

^{K}

^{e}−E*K**'*

*k**B**T* *d E*^{'}* _{K}* (2.16)

[Kett96]. For an atom to remain untrapped in the vicinity of the dipole trap, it has to have kinetic
energy bigger than the depth of the dipole trap, so it’s energy has to be bigger than (η +3/2)k*B**T. If a *
collision of two atoms is to end up with one of them trapped in the dipole potential, then one of the
Fig. I.2.3: The dependence on the final atom number (a) and final temperature (b) on the dipole trap’s waist
during the loading process. The atoms in the reservoir have an initial temperature of 150 µK and the dipole
laser total power is 100 W.

atoms has to have kinetic energy smaller than ηk*B**T leaving the other atom with energy that equals to *
*(η +3)k**B**T. The probability for such an event to occur equals to*

*p** _{t}*=

*p* 3

*k*

_{B}*T*

3/2k_{B}*T*≈0.5 (2.17)

The reverse process, which is the ejection of a trapped atom after a collision is estimated with similar arguments to be ~0.15 [Comp06]. Since such a probability is considered to be very small the process is ignored. Actually, the process of collision between an atom of the reservoir and an atom trapped in the optical potential ends up with the trapping of the reservoir atom as well, with a high probability. This process is again omitted in order to keep this simulation conservative.

Once the probability of each type of collision is calculated, we can consider the atomic flux
in the trap region. We note the number of atoms in the dimple region with a common magnitude N*d*.
In the initial times, it is the atoms from the reservoir that contribute to N*d* as N*d** = N**i** (r**d** / l)** ^{3}* and in
the end of the loading process it is the atoms trapped in the dipole potential that contribute to it N

*d*

*=*

*N*

*f*

*>> N*

*i*

*(r*

*d*

*/ l)*

*, where r*

^{3}*d*is the radius of the dipole and l the radius of the magnetic trap. The radius

*r*

*d*can be approximately connected to the waist w

*0*as r

*d*

*= 2*

^{1/2}*w*

*0*while l is considered to be the radius in which 80% of the magnetically trapped atoms are found. The two characteristic times on which the loading depends on, are the oscillation period of the atoms in the magnetic trap t

*osc*and the temporal distance t

*coll*between two atomic collisions

*t**osc*=

###

^{k}^{m l}

^{B}

^{T}^{2}

^{ and }

^{t}

^{coll}^{≈}

^{n}

^{i}^{ }

^{1}

^{i}^{(2.18)}

where n*i* the density, σ the elastic collision cross section and υ*i* the velocity of the atoms. We assume
not to be in the hydrodynamic regime, which means that t*coll** > t**osc*. An atom oscillates in the
reservoir on average t*coll** / t**osc* times before colliding with an atom, and the probability that an atom
ends up trapped after this collision is given by p*d** ~ p**t** r**d**(n**d *σ) t*coll** / t**osc* with n*d* being the density in the
dimple region. Thus, the number of atoms finally trapped, can be estimated as the product of the
number of atoms passing the dimple area with this probability, leading to the differential equation
*dN**d**/dt ~ N**d** σ N**i**/2t**osc**l** ^{2}*. As a final point of the loading we can consider the point where N

*f*

*= n*

*d,0*

*r*

*d*

*3*. With these, the loading time can be estimated to be

*t** _{load}*= 2t

_{osc}*l*

^{2}

*N** _{i}*ln

*N*

_{f}*N*_{i}*r*_{d}^{3}/*l*^{3}≈2t* _{coll}*ln

*n*

_{f}*n** _{i}* (2.19)

The losses that are related to the loading process are the two and three body inelastic losses
as well as the Majorana losses. In absence of resonant effects like Feshbach resonances, the heating
associated with two body inelastic collisions is proportional to B* ^{1/2 }*(due to the dependence of the
energy of states on the magnetic field). Thus, the use of a quadrupole field for the realization of the
magnetic trap which plays here the role of the reservoir is an advantage, since it provides a large
area in its center where the field has small values. The two body inelastic rate for Cs is K

*2*

*= 410*

^{-11 }*B*

^{1/2}*mT(µK)*

*cm*

^{ -0.78}^{3}s

^{-1}[Gue98]and the equation for the rate can be written as

*N*˙_{2}=*K*_{2}*n*_{d}^{2}*r*_{d}^{3} giving for the lost atoms *N*_{2,}* _{f}*≈t

_{coll}*K*

_{2}

*n*

_{d}^{2}

*r*

_{d}^{3}(2.20) Thus the lost fraction is N

*2,f*

*/ N*

*f*

*˜*

*t*

*coll*

*K*

*2*

*n*

*f*. Similarly for the three body inelastic loss ( rate ƒ ~ L

*3*

*n**d**2*), the lost fraction will be N*3,f** / N**f**˜ **(2/3) t**coll** L**3** n**f**2*. As noted before, the Majorana loss consists of
spin flips which occur when atoms pass from an area of small magnetic field, the Majorana sphere,
with radius r*Maj** = ( u**d** /µ**B** ) which is in the order of 1µm typically [Comp06], where u**d* is the

With these considerations for the dynamics of the loading process we can have an estimate
of the loading time and the number lost atoms. The final atom number is calculated with the
thermodynamic arguments that do not take into account the dynamic losses; nevertheless, a more
realistic final number of atoms can be estimated as N*f **– N**loss**. In Fig. I.2.4 we give an estimate for the *
loading time, the lost atom number and the final number of atoms, for a magnetic trap which
contains ~ 10^{8} atoms in a temperature of 150 µK and for a laser power of 100 Watts.